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On M-functions associated with modular forms
Philippe Lebacque, Alexey Zykin
To cite this version:
Philippe Lebacque, Alexey Zykin. On M-functions associated with modular forms. 2016. �hal- 01475616�
ON M-FUNCTIONS ASSOCIATED WITH MODULAR FORMS
PHILIPPE LEBACQUE AND ALEXEY ZYKIN
Abstract. Let f be a primitive cusp form of weightk and level N, let χ be a Dirichlet character of conductor coprime with N, and let L(f ⊗χ, s) denote either logL(f ⊗χ, s) or (L′/L)(f⊗χ, s).In this article we study the distribution of the values ofLwhen either χ or f vary. First, for a quasi-character ψ: C → C× we find the limit for the average Avgχψ(L(f ⊗χ, s)),whenf is fixed andχvaries through the set of characters with prime conductor that tends to infinity. Second, we prove an equidistribution result for the values of L(f ⊗χ, s) by establishing analytic properties of the above limit function. Third, we study the limit of the harmonic average Avghfψ(L(f, s)), when f runs through the set of primitive cusp forms of given weightk and levelN → ∞.Most of the results are obtained conditionally on the Generalized Riemann Hypothesis forL(f⊗χ, s).
1. Introduction
1.1. Some history. The study of the distribution of values ofL-functions is a classical topic in number theory. In the first half of 20th century Bohr, Jessen, Wintner, etc. intiated a study of the distribution of the values of the logarithm logζ(s) and the logarithmic derivative (ζ′/ζ)(s) of the Riemann zeta-function, when Res=σ > 12 is fixed and Ims =τ ∈Rvaries ([1], [2],[18], [19]). This was later generalized to L-functions of cusp forms and Dedekind zeta-functions by Matsumoto ([23], [24], [25]).
In the last decade Y. Ihara in [5] proposed a novel view on the problem by studying other families of L-functions. His initial motivation was to investigate the properties of the Euler–Kronecker constant γK of a global field K, which was defined by him in [4] to be the constant term of the Laurent series expansion of the logarithmic derivative of the Dedekind zeta function of K, ζK′ (s)/ζK(s). The study of L′(1, χ)/L(1, χ) initiated in [13] grew out to give a whole range of beautiful results on the value distribution of L′/L and logL.
Given a global field K, i.e. a finite extension of Q or of Fq(t), and a family of characters χ of K Ihara considered in [5] the distribution of L′(s, χ)/L(s, χ) in the following cases:
(A) K is Q, a quadratic extension of Q or a function field over Fq, χ are Dirichlet characters on K;
(B) K is a number field with at least two archimedean primes, and χ are normalized unramified Gr¨ossencharacters;
(C) K =Q and χ=χt, t∈R defined by χt(p) =p−it. The equidistribution results of the type
Avg′χΦ
L′(s, χ) L(s, χ)
= Z
C
Mσ(w)Φ(w)|dw|, (1)
2010Mathematics Subject Classification. Primary 11F11, Secondary 11M41.
Key words and phrases. L-function, cuspidal newforms, value-distribution, density function.
The two authors were partially supported by ANR Globes ANR-12-JS01-0007-01 and the second author by the Russian Academic Excellence Project ’5-100’.
1
(with a suitably defined average in each of the above cases) were proven for σ = Res > 1 for number fields, and for σ > 3/4 for function fields, under significant restrictions on the test function Φ. The function field case was treated once again in [9] by Y. Ihara and K. Matsumoto, with both the assumptions on Φ and on σ having been relaxed (Φ of at most polynomial growth and σ > 1/2 respectively). The most general results in the direction of the case (A) were established in [11] conditionally under the Generalized Riemann Hypothesis (GRH) in the number field case and unconditionally in the function field case (the Weil’s Riemann hypothesis being valid) for both families L′(s, χ)/L(s, χ) and logL(s, χ). For Res > 12 Ihara and Matsumoto prove that
AvgχΦ
L′(s, χ) L(s, χ)
= Z
C
Mσ(w)Φ(w)|dw|, AvgχΦ(logL(s, χ)) = Z
CMσ(w)Φ(w)|dw|, for continuous test functions Φ of at most exponential growth. Note that Avg′ in (1) is different from the one used in the latter paper, since extra averaging over conductors is assumed in the former case, the resulting statements being weaker.
Unconditional results for a more restrictive class of Φ (bounded continous functions), and with extra averaging over the conductor Avg′, but still for Res > 12 were established in [10]
and [12] in the log and log′ cases respectively in the situations (A, K =Q) and (C).
The above results give rise to the density functionsMσ(z) and a related function ˜Ms(z1, z2) (which is the inverse Fourier transform of Mσ, when z2 = ¯z1, s = σ ∈ R) both in the log and log′ cases. Under optimal circumstances (though it is very far from being known unconditionally in all cases) we have
Mσ(z) = Avgχδz(L(χ, s)), M˜σ(z1, z2) = Avgχψz1,z2(L(χ, s)),
where L(s, χ) is either L′(s, χ)/L(s, χ) or logL(s, χ), δz is the Dirac delta function, and ψz1,z2(w) = exp 2i(z1w¯+z2w)
is a quasi-character.
The functions M and ˜M turn out to have some remarkable properties that can be estab- lished unconditionally. For example, ˜M has an Euler product expansion, an analytic con- tinuation to the left of Res >1/2, its zeroes and the “Plancherel volume”R
C|M˜σ(z,z)¯ |2|dz| are interesting objects to investigate. We refer to [6], [7] for an in-depth study ofM and ˜M , as well as to the survey [8] for a thorough discussion of the above topics.
In a recent paper by M. Mourtada and K. Murty [27] averages over quadratic characters were considered. Using the methods from [11], they establish an equidistribution result conditional on GRH. Note that in their case the values taken by theL-functions are real. In this respect the situation is similar to the one considered by us in§5 in case we assume that s is real.
Finally, let us quote a still more recent preprint by K. Matsumoto and Y. Umegaki [26]
that treats similar questions for differences of logarithms of two symmetric powerL-functions under the assumption of the GRH. Their approach is based on [10] rather than on [11], though the employed techniques are remarkably close to the ones we apply in §5. The results of Matsumoto and Umegaki are complementary to ours, since the case of Sym1f =f,which is the main subject of our paper, could not be treated in [26].
1.2. Main results. In this article, we generalize to the case of modular forms the methods of Ihara and Matsumoto to understand the average values ofL-functions of Dirichlet characters over global fields.
2
Our results are obtained in two different setting. First, we consider the case of a fixed modular form, while averaging with respect to its twists by Dirichlet characters. Our results in this setting are fairly complete, though sometimes conditional on GRH. Second, we con- sider averages with respect to primitive forms of given weight and level, when the level goes to infinity.
Let us formulate our main results. A more thorough presentation of the corresponding notation can found in §2 and in the corresponding sections.
LetBk(N) denote the set of primitive cusp forms of weightk and level N,let f ∈Bk(N), and let χ be a Dirichlet character of conductor m coprime with N. Define L(f⊗χ, s) to be either (L′/L)(f⊗χ, s) or logL(f⊗χ, s),put g(f⊗χ, s, z) = exp iz2L(f, s)
. We introduce lz(n) to be the coefficients of the Dirichlet series expansiong(f⊗χ, s, z) = P
n≥1
lz(n)n−s.Using the relations between the coefficients of the Dirichlet series expansionL(f, s) = P
n≥1
ηf(n)n−s, one can writelz(n) = P
x≥1
cNz,x(n)ηf(x),wherecNz,x(n) depend only on the levelN.Putcz,x(n) = c1z,x(n).
In what follows, the expressions of the form f ≪ g, g ≫f, and f =O(g) all denote that
|f| ≤ c|g|, where c is a positive constant. The dependence of the constant on additional parameters will be explicitly indicated (in the form ≪ǫ,δ,... or Oǫ,δ,...), if it is not stipulated otherwise in the text. We denote by vp(n) thep-adic valuation ofn, writing as wellpk kn if vp(n) =k.We also use the notation := or =: meaning that the corresponding object to the left or to the right of the equality respectively is defined in this way.
Our main results are as follows.
Theorem (Theorem 3.1). Assume that m is a prime number and let Γm denote the group of Dirichlet characters modulo m. Let 0 < ǫ < 12 and T, R > 0. Let s = σ +it belong to the domain σ ≥ ǫ+ 12, |t| ≤ T, let z and z′ be inside the disk DR = {z | |z| ≤ R}. Then, assuming the Generalized Riemann Hypothesis (GRH) for L(f⊗χ, s), we have
mlim→∞
1
|Γm| X
χ∈Γm
g(f⊗χ, s, z)g(f ⊗χ, s, z′) =X
n≥1
lz(n)lz′(n)n−2σ =: ˜Mσ(−z, z¯ ′).
Theorem (Theorem 4.1). Let Res = σ > 12 and let m run over prime numbers. Let Φ be either a continuous function on C with at most exponential growth, or the characteristic function of a bounded subset of C or of a complement of a bounded subset of C. Define Mσ
as the inverse Fourier transform of M˜σ(z,z).¯ Then under GRH for L(f ⊗χ, s) we have
mlim→∞
1
|Γm| X
χ∈Γm
Φ(L(f⊗χ, s)) = Z
C
Mσ(w)Φ(w)|dw|.
Theorem (Theorem 5.1). Assume that N is a prime number and that k is fixed. Let 0 <
ǫ < 12 and T, R >0. Let s=σ+it belong to the domain σ≥ǫ+ 12, |t| ≤T, and z and z′ to the disc DR of radius R. Then, assuming GRH for L(f, s), we have
Nlim→+∞
X
f∈Bk(N)
ω(f)g(f, s, z)g(f, s, z′) = X
n,m∈N
n−s¯m−sX
x≥1
cz,x(n)cz′,x(m), where ω(f) are the harmonic weights defined in §5.
3
Finally, let us describe the structure of the paper. In §2 we introduce the notation and some technical lemmas to be used throughout the paper. The §3 is devoted to the proof of Theorem 3.1 on the mean values of the logarithms and logarithmic derivatives ofL-functions obtained by taking averages over the twists of a given primitive modular form. Using GRH, we deduce it from Ihara and Matsumoto’s results. In§4 we study unconditionally the analytic properties of M and ˜M functions in the above setting. We then prove an equidistribution result (Theorem 4.1), which is, once again, conditional on GRH. In §5, we consider the average over primitive forms of given weight k and level N, when N → ∞, establishing under GRH Theorem 5.1. The orthogonality of characters is replaced by the Petersson formula in this case, which obviously makes the proofs trickier. Finally, open questions, remarks and further research directions are discussed in §6.
Acknowledgements.
We would like to thank Yasutaka Ihara for helpful discussions. The first author would like to express his gratitude to the INRIA team GRACE for an inspirational atmosphere accompanying his stay, during which a large part of this work was done.
2. Notation
The goal of this section is to introduce the notation necessary to state our main results.
We also prove some auxiliary estimates to be used throughout the paper.
2.1. The g-functions. LetN,k be two integers. We denote bySk(N) the set of cusp forms of weight k and level N, and by Sknew(N) the set of new forms. For f ∈ Sk(N) we write f(z) = P∞
n=1
ηf(n)n(k−1)/2e(nz) for its Fourier expansion at the cusp ∞, with the standard notation e(nz) =e2πinz.
LetBk(N) denote the set of primitive forms of weightk and level N, i.e. the set offnor = f /ηf(1) where f runs through an orthogonal basis of Sknew(N) consisting of eigenvectors of all Hecke operatorsTn, so that the Fourier coefficients of the elements ofBk(N) are the same as their Hecke eigenvalues. Note that for a primitive formf ∈Bk(N) all its coefficientsηf(n) are real.
The L-function of a primitive form f ∈Bk(N) is defined as the Dirichlet series L(f, s) = P∞
n=0
ηf(n)n−s. The series converges absolutely for Res > 1, however, L(f, s) can be analyti- cally continued to an entire function on C.It admits the Euler product expansion:
L(f, s) =Y
p
Lp(f, s), where, for any prime number p,
Lp(f, s) =
((1−ηf(p)p−s+p−2s)−1 if (p, N) = 1, (1−ηf(p)p−s)−1 if p|N.
By the results of Deligne, these local factors can be written as follows ([14, Chapter 6] or [20, Chapter IX, §7]):
Lp(f, s) = 1−αf(p)p−s−1
1−βf(p)p−s−1
, (2)
4
where
|αf(p)|= 1, βf(p) =αf(p)−1 if (p, N) = 1,
αf(p) =±p−12, βf(p) = 0 if pkN (that isp|N and p2 ∤N), αf(p) =βf(p) = 0 if p2 |N.
We are interested in the two functions
g(f, s, z) = exp iz
2
L′(f, s) L(f, s)
, G(f, s, z) = exp
iz
2 logL(f, s)
.
Define hn(x) and Hn(x) as the coefficients of the following generating functions:
exp xt
1−t
=
+∞
X
n=0
hn(x)tn,
exp(−xlog(1−t)) =
+∞
X
n=0
Hn(x)tn,
or, equivalently (cf. [11, §1.2]), as the functions given byh0(x) =H0(x) = 1 and, for n≥1,
hn(x) =
n
X
r=0
1 r!
n−1 r−1
xr, Hn(x) = 1
n!x(x+ 1). . .(x+n−1).
As we have
iz 2
L′(f, s)
L(f, s) =−iz 2
X
p
αf(p)p−slogp
1−αf(p)p−s + βf(p)p−slogp 1−βf(p)p−s ,
5
we can write (using the standard convention that, in the case when βf(p) = 0, we put βf(p)n = 0, if n >0, and βf(p)0 = 1):
g(f, s, z) = exp iz
2
L′(f, s) L(f, s)
=
=Y
p
exp
αf(p)p−s
1−αf(p)p−s · −izlogp 2
exp
βf(p)p−s
1−βf(p)p−s · −izlogp 2
=
=Y
p
X
n
hn
−iz 2 logp
αf(p)np−ns
! X
n
hn
−iz 2 logp
βf(p)np−ns
!
=
=Y
p
+∞
X
n=0 n
X
r=0
hr
−iz 2 logp
hn−r
−iz 2 logp
αf(p)rβf(p)n−rp−ns
!
=
=Y
p∤N +∞
X
n=0 n
X
r=0
hr
−iz 2 logp
hn−r
−iz 2 logp
αf(p)2r−np−ns
!
·
·Y
pkN +∞
X
n=0
hn
−iz 2 logp
αf(p)np−ns=:Y
p +∞
X
n=0
λz(pn)p−ns. In a similar way we get:
G(f, s, z) = exp iz
2 logL(f, s)
=
=Y
p
exp
−iz
2 log(1−αp(f)p−s)
exp
−iz
2 log(1−βp(f)p−s)
=
=Y
p∤N +∞
X
n=0 n
X
r=0
Hr
iz 2
Hn−r
iz 2
αf(p)2r−np−ns
! Y
pkN +∞
X
n=0
Hn
iz 2
αf(p)np−ns
=:Y
p +∞
X
n=0
Λz(pn)p−ns.
We extend multiplicatively λz and Λz toN so that we can write:
g(f, s, z) = X
n≥1
λz(n)n−s, G(f, s, z) =X
n≥1
Λz(n)n−s.
We will use the notationLfor L′(f, s)
L(f, s) or logL(f, s),gforg orG,hz(pn) forhn −iz2 logp or Hn iz
2
, and l for λ or Λ depending on the case we consider. Thus, we can write in a uniform way:
6
g(f, s, z) = exp iz
2L(f, s)
=X
n≥1
lz(n)n−s =Y
p +∞
X
n=0
lz(pn)p−ns=
=Y
p∤N +∞
X
n=0 n
X
r=0
hz(pr)hz(pn−r)αf(p)2r−np−ns
! Y
pkN +∞
X
n=0
hz(pn)αf(p)np−ns.
The coefficients lz(n) will be used to define the ˜M-functions in the case of averages over twists of modular forms by Dirichlet characters.
2.2. The coefficients lz(n) and cz,x(n). In this subsection we will find a more explicit expression forlz(n).For p∤N we will use the formula (see [30, (3.5)])
ηf(pr) = αf(p)r+1−βf(p)r+1 αf(p)−βf(p) ,
which easily follows from (2). Taking into account that βf(p) = ¯αf(p), we have for r≥2 ηf(pr) = αf(p)r+1−αf(p)r+1
αf(p)−αf(p) =
r
X
i=0
αf(p)iαf(p)r−i=
r
X
i=0
αf(p)r−2i =
=αf(p)r+αf(p)r+
r−1
X
i=1
αf(p)r−2i =αf(p)r+αf(p)r+
r−2
X
i=0
αf(p)r−2i−2 =
=αf(p)r+αf(p)r+ηf(pr−2).
The above formula also holds for r = 1 if we put ηf(p−1) = 0. From this we deduce that αf(p)r+βf(p)r=ηf(pr)−ηf(pr−2).
Using the previous formula, we can write lz(pr) =
r
X
a=0
hz(pa)hz(pr−a)αf(p)2a−r
=hz(pr2)2+
⌊r−12 ⌋
X
a=0
hz(pa)hz(pr−a) αf(p)r−2a+αf(p)2a−r
=hz(pr2)2+
⌊r−12 ⌋
X
a=0
hz(pa)hz(pr−a) ηf(pr−2a)−ηf(pr−2a−2)
=hz(pr2)2−hz(pr2−1)hz(pr2+1) +
⌊r−12 ⌋
X
a=0
(hz(pa)hz(pr−a)−hz(pa−1)hz(pr−a+1))ηf(pr−2a)
=
⌊r2⌋
X
a=0
(hz(pa)hz(pr−a)−hz(pa−1)hz(pr−a+1))ηf(pr−2a),
where we put hz(pr2) =hz(pr2−1) = 0,if r is odd, andhz(pa) = 0, if a <0.
7
When p|N we have
lz(pr) =hz(pr)αf(p)r =hz(pr)ηf(p)r=hz(pr)ηf(pr).
Denoting by P the set of prime numbers, for n= Q
p∈P
pvp(n) put
IN(n) ={m ∈N|vp(m)≡vp(n) mod 2 for p∈ P, vp(n) = vp(m) ifp|N} and
JN(n) ={m ∈IN(n)|vp(m)≤vp(n) for all p∈ P}.
Note the following easy estimate ([3, Theorem 315]) in which τ(n) is the number of divisors of n:
|JN(n)|=Y
p|n
vp(n) 2
+ 1
≤τ(n)≪ǫnǫ. (3) The previous computations may be summarized as follows:
lz(pr) = X
x∈JN(pr)
cNz,x(pr)ηf(x), where
cNz,pa(pr) =
hz(pr−a2 )hz(pr+a2 )−hz(pr−a2 −1)hz(pr+a2 +1), if p∤N and r≡amod 2,
hz(pr), if p|N and r=a,
0, otherwise.
We have lz(n) = Y
p|n
lz(pvp(n)) and ηf(n)ηf(m) =ηf(nm) if (n, m) = 1,thus
lz(n) =Y
p|n
X
x∈JN(pvp(n))
cNz,x(pvp(n))ηf(x)
= X
x∈JN(n)
cNz,x(n)ηf(x), with
cNz,x(n) =Y
p|n
cNz,pvp(x)(pvp(n)).
Note that the coefficients cNz,x(n), IN(n), and JN(n) depend only on the level N and not directly on the modular formf.Let us also define I(n) =I1(n), J(n) =J1(n),and cz,x(n) = c1z,x(n). They will employed in the statement of Theorem 5.1, which is our main result on averages over the set of primitive forms Bk(N).
LetB(a, R) = {z∈C| |z−a|< R}denote the open disc of radiusR and centera∈C,let B(a, R) be the corresponding closed disc. We also putDR=B(0, R).The following estimate is used throughout the paper.
Lemma 2.1. For any ǫ >0 and z ∈ DR we have |cNz,x(n)| ≪ǫ,R nǫ and |lz(n)| ≪ǫ,R nǫ. Proof. To see this, recall ([11, 3.1.2]) that for any primep
Hr
iz 2
≤Hr
|z| 2
≤hr
|z| 2
≤hr(|z|logp)
8
and
hr
−iz 2 logp
≤hr(|z|logp)≤exp 2p
r|z|logp , thus in both cases |hz(pr)| ≤ exp
2p
r|z|logp
. Using the concavity of the function √ x, we see that
|cNz,x(pr)| ≤e2√r−a
2 |z|logpe2√r+a
2 |z|logp+e2√
(r−a2 −1)|z|logpe2√
(r+a2 +1)|z|logp
≤e2
√|z|logp√r−a
2 +√r+a
2
+e2
√|z|logp√r−a
2 −1+√r+a
2 +1
≤e2√
|z|logp√
2r+e2√
|z|logp√
2r ≤2e2√
2r|z|logp.
when p ∤ N. The above estimates on hz(pr) also imply the same bound on cNz,x(pr) when p|N.
Now, denoting by ω(n) the number of distinct prime divisors of n and using once again the concavity of√
x, for n= Q
p∈P
pvp(n) we have
log|cNz,x(n)| ≤X
p|n
(log 2 + q
vp(n) logp√
8R)≪R
X
p|n
q
vp(n) logp
√8R
≪R sX
p|n
vp(n) logpp
ω(n)≪R s
logn 2 + log logn
plogn,
since by [5, Sublemma 3.10.5] (which is classical in the case ofN) we have ω(n)≪ logn
2 + log logn. (4)
We thus conclude that |cNz,x(n)| ≪ǫ,Rnǫ.
As for the second statement, we notice that the estimate (3) together with Deligne bound
|ηf(n)| ≤τ(n)≪ǫnǫ imply lz(n)≪ǫ |JN(n)| ·nǫ·τ(n)≪ǫ n3ǫ. We conclude the section by the following trivial but useful lemma.
Lemma 2.2. We have lz(n) =l−z¯(n), and cNz,x(n) =cN−z,x¯ (n).
Proof. The eigenvalues ηf(n) are all real, so the L-functions L(f, s) have Dirichlet series with real coefficients. Thus the statement of the lemma follows from the definition of the
coefficients lz(n), and cNz,x(n).
3. Average on twists
This section is devoted to the proof of an averaging result for twists of a given primitive form. It is to a large extent based on the work of Ihara and Matsumoto [11], which provides a general setting for the problem we consider.
9
3.1. Setting. Let us fix a primitive cusp form f ∈ Bk(N) of weight k and level N. Let χ: (Z/mZ)× →C× be a primitive character modm, where (m, N) = 1.It is known (see [15, Prop. 14.19 and Prop. 14.20]) that f ⊗χ is a primitive form of weight k, level Nm2, and nebentypus χ2. We consider the twisted L-function given by
L(f⊗χ, s) =Y
p
Lp(f⊗χ, s), where the local factors are defined as follows:
Lp(f ⊗χ, s) = 1−αf(p)χ(p)p−s−1
1−βf(p)χ(p)p−s−1
,
with the notation of §2. It is an L-function of degree 2 and conductor Nm2, entire and polynomially bounded in vertical strips. After multiplication by the gamma factor
γk(s) =√
π23−k2 (2π)−sΓ
s+k−1 2
,
it satisfies a functional equation [15, §5.11]. Its analytic conductor q(f⊗χ, s) is defined as follows:
q(f ⊗χ, s) =Nm2
s+k−1 2
+ 3
s+k+ 1 2
+ 3
≤Nm2(|s|+k+ 3)2. Just as in§2 we use the following notation for modular forms with nebentypus:
g(f ⊗χ, s, z) = exp iz
2
L′(f ⊗χ, s) L(f ⊗χ, s)
, G(f⊗χ, s, z) = exp
iz
2 logL(f⊗χ, s)
. We also write g(f ⊗χ, s, z) to denote either of the above two functions.
If G is a function on a finite group K, let Avgχ∈KG(χ) denote the usual average
|K|−1P
χ∈KG(χ).
3.2. TheM˜-function. We would like to understand the average over all Dirichlet characters modmof the functions g(f⊗χ, s, z), whenm runs through large prime numbers. Ihara and Matsumoto’s results apply in this case and we get the following theorem.
Theorem 3.1. Assume that m is a prime number. Let Γm denote the group of Dirichlet characters modulo m. Let 0 < ǫ < 12 and T, R > 0. Let s = σ +it belong to the domain σ ≥ ǫ + 12, |t| ≤ T, let z and z′ be inside the disk DR. Then, assuming the Generalized Riemann Hypothesis (GRH) for L(f ⊗χ, s), in the notation of §2 we have
Avg
χ∈Γm
g(f ⊗χ, s, z)g(f⊗χ, s, z′)
− X
(n,m)=1
lz(n)lz′(n)n−2σ ≪ǫ,R,T,f m−ǫ2. (5) Moreover,
mlim→∞Avg
χ∈Γm
g(f⊗χ, s, z)g(f ⊗χ, s, z′)
=X
n≥1
lz(n)lz′(n)n−2σ.
10
Proof. We notice that g(f ⊗χ, s, z) = P
n≥1
lz(n)χ(n)n−s, where lz(n) are the coefficients of g(f, s, z). We thus can deduce the theorem from [11, Theorem 1]. We can pass to the situation treated in [11] by omitting the summand corresponding to the trivial character χ0
since in our case all the g(f ⊗χ, s, z) are holomorphic for Res > 12. Thus, it is enough to prove that the family l|z|≤R is uniformly admissible in the sense of Ihara and Matsumoto.
First of all, the property (A1), asserting that l|z|≤R(n)≪ǫnǫ,follows from Lemma 2.1.
The property (A2) states thatg(f⊗χ, s, z) extend to holomorphic functions on Res > 12 for any non trivial χ, which is true under GRH.
The property (A3) will be proven in the following lemma that will be used again in §5.
Lemma 3.2. Let f be a primitive form of weight N, and let χ be a primitive Dirichlet character of conductor m coprime with N. Then, assuming GRH for L(f ⊗χ, s), we have for Res≥ 12 +ǫ:
max(0,log|g(f⊗χ, s, z)|)≪ǫ,R ℓ(t)1−2ǫℓ(mNk)1−2ǫ, where ℓ(x) = log(|x|+ 2), t= Ims.
Proof of the Lemma. First, the following estimates hold ([15, Theorems 5.17 and 5.19]) for any s with 12 <Res=σ ≤ 54 :
−L′(f ⊗χ, s) L(f ⊗χ, s) =O
1
2σ−1(logq(f ⊗χ, s))2−2σ+ log logq(f ⊗χ, s)
, and
logL(f ⊗χ, s) =O
(logq(f ⊗χ, s))2−2σ
(2σ−1) log logq(f⊗χ, s) + log logq(f⊗χ, s)
, the implied constants being absolute.
Next, for the same range of s we have
logq(f⊗χ, s)≪log(mNk) + log(|t|+ 2)≪ℓ(mNk) +ℓ(t).
Thus we see that
log|g(f⊗χ, s, z)|= log
exp iz
2L(f⊗χ, s)
= Re iz
2L(f ⊗χ, s)
≪R|L(f⊗χ, s)|, so
max(0,log|g(f⊗χ, s, z)|)≪ǫ,R ℓ(t)1−2ǫℓ(Nmk)1−2ǫ.
Ifσ ≥ 54 a much simpler estimate suffices. Indeed, using the fact that [15, (5.25)]
−L′(f, s)
L(f, s) =X
n
Λf(n)
ns and logL(f, s) =−X
n
Λf(n) nslogn,
with Λf(n) supported on prime powers and Λf(pn) = (αf(p)n+βf(p)n) logp, we see that both L′(f, s)
L(f, s) and logL(f⊗χ, s) are bounded by an absolute constant. Thus the conclusion
of the lemma still holds in this case.
Thus Ihara and Matsumoto’s property (A3) is established (with a stronger bound than required), since in our caseN andkare fixed. So, the family we consider is indeed uniformly
admissible.
11
Remark 3.3. The estimate (5) should still be true if we omit the condition on m to be prime. To prove it one establishes an analogue of Lemma 3.2, replacing χwith the primitive character by which it is induced and estimating the bad factors of theL-function (with some additional work required when m is not coprime with N). Then one uses once again [11, Theorem 1], in which the first inequality is true without any restriction on the conductor.
Remark 3.4. The theorem should hold unconditionally forσ = Res >1 by orthogonality of characters, all the series being absolutely convergent in this domain.
As a direct consequence, we obtain the following result on averages of the values ofg. Put M˜s(z1, z2) =
X∞
n=1
lz1(n)lz2(n)n−2s.
Because of Lemma 2.1, the series converges uniformly and absolutely on Res ≥ 12 +ε,
|z1|,|z2| ≤R, defining a holomorphic function of s, z1, z2 for Res > 12. Put ψz1,z2(w) = exp
i
2(z1w+z2w)
.
Corollary 3.5. Let m run over prime numbers. Then, assuming GRH,
mlim→∞Avg
χ∈Γm
ψz1,z2(L(f⊗χ, s)) = ˜Mσ(z1, z2).
Proof. By definition, we have :
ψz1,z2(L(f⊗χ, s)) = exp i
2z1L(f ⊗χ, s)
exp i
2z2L(f ⊗χ, s)
=g(f⊗χ, s,−z¯1)g(f⊗χ, s, z2).
By Theorem 3.1 we get
mlim→∞Avg
χ∈Γm
ψz1,z2(L(f⊗χ, s)) = X
n≥1
l−z¯1(n)lz2(n)n−2σ.
Lemma 2.2 implies that l−¯z(n) =lz(n),so the corollary is proven.
4. The distribution of L-values for twists
Our next result concerns the distribution of the values of logarithmic derivatives and logarithms ofL-functions of twists of a fixed modular formf. In this section the dependence onf in≪ will be omitted.
Recall that we have defined
M˜s(z1, z2) =
∞
X
n=1
lz1(n)lz2(n)n−2s,
the corresponding series being absolutely and uniformly convergent on Res ≥ 12 +ǫ,|z1| ≤ R,|z2| ≤R. Forσ ∈R, we put ˜Mσ(z) = ˜Mσ(z,z).¯
Define the family of additive characters ψz1,z2(w) = exp
i
2(z1w+z2w)
.
12
We also letψz(w) =ψz,¯z(w) = exp(iRe(zw)).¯ Recall that the Fourier transform ofφ:C→C, φ∈L1 is defined as
Fφ(z) = Z
C
φ(w)ψz(w)|dw|= 1 2π
Z
C
φ(w)eiRe(zw)¯ |dw|= 1 2π
Z
R2
φ(w)ei(xx′+yy′)dxdy, where |dw|= 1
2πdxdy, x= Rew, y= Imw, x′ = Rez, y′ = Imz.
The goal is to prove the following equidistribution result, which is an analogue of [11, Theorem 4].
Theorem 4.1. Let Res = σ > 12 and let m run over prime numbers. Let Φ be either a continuous function on C with at most exponential growth, that is Φ(w) ≪ ea|w| for some a >0,or the characteristic function of a bounded subset ofCor of a complement of a bounded subset ofC.DefineMσ as the inverse Fourier transform ofM˜σ(z), Mσ(z) =FM˜σ(−z).Then under GRH for L(f ⊗χ, s) we have
mlim→∞Avg
χ∈Γm
Φ(L(f ⊗χ, s)) = Z
C
Mσ(w)Φ(w)|dw|. (6)
Remark 4.2. The above theorem should hold unconditionally for anyσ > 1 and any contin- uous function Φ on C,by virtue of Remarks 3.4 and (iv) of Corollary 4.12.
To prove this theorem we first construct the local M and ˜M-functions and establish their properties. We then obtain a convergence result for partialM-functions Ms,P for finite sets of primes P to a global function M. This allows us to prove some crucial estimates for the growth of M. Finally, we deduce the global result using corollary 3.5. Our approach is strongly influenced by that of Ihara and Matsumoto, the main ingredients being inspired by the results of Jessen and Wintner [18] that we have to adapt to our situation.
All the results below, except from the proof of Theorem 4.1 itself, do not depend on GRH.
4.1. The functions Ms,P and M˜s,P. Let Res=σ > 0.Define the functions on Tp =C1 = {t∈C | |t|= 1}by
gs,p(t) = −(logp)α(p)p−st
1−α(p)p−st + −(logp)β(p)p−st 1−β(p)p−st , and
Gs,p(t) =−log(1−α(p)p−st)−log(1−β(p)p−st).
As before, we let gs,p denote either gs,p orGs,p,depending on the case we consider. We note that the local factor of the L-function is 1 oncep2 |N,so we can omit such primes from our considerations.
Denote by fp(z) the expression
−(logp)α(p)z
1−α(p)z +−(logp)β(p)z
1−β(p)z or −log(1−α(p)z)−log(1−β(p)z).
in the log′ and log case respectively. Note that if p ∤N, fp(z) =−logp· ηf(p)z−2z2 1−ηf(p)z+z2 or
−log(1−ηf(p)z +z2) respectively. The functions fp(z) are holomorphic in the open disc
|z|<1. We obviously have gs,p(t) =fp(p−st).
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For a prime number p, let Tp = C1 be equipped with the normalized Haar measure d×t = dt
2πit. If P is a finite set of primes, we let TP = Q
p∈P
Tp and we denote by d×tP the normalized Haar measure on TP. Put alsogs,P =X
p∈P
gs,p. We introduce the local factors ˜Ms,p(z1, z2) via
M˜s,p(z1, z2) =
+∞
X
r=0
lz1(pr)lz2(pr)p−2rs. (7) The series is absolutely and uniformly convergent on compacts in Res > 0 by Lemma 2.1.
Put ˜Ms,P(z1, z2) = Q
p∈P
M˜s,p(z1, z2). We also define ˜Mσ,p(z) = ˜Mσ,p(z,z),¯ and ˜Mσ,P(z) = M˜σ,P(z,z).¯
Lemma 4.3. (i) The function M˜s,P(z1, z2) is entire in z1, z2. (ii) We have
M˜s,p(z1, z2) = Z
C1
exp i
2(z1gs,p(t−1) +z2gs,p(t))
d×t.
In particular, M˜σ,p(z1, z2) =
Z
C1
ψz1,z2(gσ,p(t))d×t, and M˜σ,p(z) = Z
C1
exp(iRe(gσ,p(t)¯z))d×t.
(iii) The “trivial” bound |M˜σ,p(z)| ≤1 holds.
Proof. (i) This is a direct corollary of the absolute and uniform convergence of the series of analytic functions (7), defining ˜Ms,p(z1, z2).
(ii) It is clear from the definitions that exp iz
2gs,p(t)
=
∞
X
r=0
lz(pr)(p−st)r. So, the state- ment is implied by the fact that ˜Ms,p is the constant term of the Fourier series expansion of exp 2i(z1gs,p(t−1) +z2gs,p(t))
.
(iii) Obviously follows from (ii).
For the sake of convenience in what follows we will identify a function on R2 with the Radon measure or the tempered distribution it defines, when the latter make sense. We will also regard the Fourier transform or the convolution products as being defined via the corresponding distributions. We refer to [18, §2,§3] for more details.
Proposition 4.4. (i) There exists a unique positive measure Mσ,P of compact support and mass 1 on C≃R2 such that
Mσ,P(Φ) = Z
TP
Φ(gs,P(tP))d×tP
for any continuous function Φ on C.
(ii) FMσ,P = ˜Mσ,P(z).
(iii) There exists a set of primes Pf of positive density such that, for all p ∈ Pf, M˜σ,p(z)≪p,σ (1 +|z|)−12.
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(iv) LetP be a set of primes. If |P∩Pf|>4, thenMσ,P admits a continuous density (still denoted by Mσ,P) which is an L1 function. The function Mσ,P satisfies Mσ,P(z) = Mσ,P(¯z)≥0.
(v) Mσ,P is of class Cr once |P ∩ Pf|>2(r+ 2).
Proof. (i) The uniqueness statement is obvious and the existence is given by the direct image measure (gs,P)∗(d×tP).The volume of an open setU ofR2 is thus given byMσ,P(U) = Vol(g−s,P1(U)), therefore Mσ,P has compact support equal to the image of gs,P and mass 1.
From the formula Ms,P(Φ) = Z
TP
Φ(gs,P(tP))d×tP, it is clear that Ms,P depends only on σ, since Haar measures on TP are invariant under multiplication by piIm(s).
(ii) From the definition of the convolution product we note that, regarded as distributions with compact support, Mσ,P =∗p∈PMσ,p.
Next,FMσ,P =F(∗p∈PMσ,P) = Q
p∈PFMσ,p.From Lemma 4.3 we see that ˜Mσ,P(z1, z2) = Mσ,P(ψz1,z2),and for the Fourier transforms of tempered distributions on C≃R2 we have
FMσ,p(φ) =Mσ,p
Z
C
ψz(w)φ(w)|dw|
= Z
Tp
Z
C
ψgs,p(t)(w)φ(w)|dw|d×t
= Z
C
Z
Tp
ψgs,p(t)(w)φ(w)d×t|dw|= Z
C
Mσ,p(ψz(w))φ(w)|dw|
= Z
C
Mσ,p(ψw(z))φ(w)|dw|= Z
C
M˜σ,p(w)φ(w)|dw|. We deduce that FMσ,P = ˜Mσ,P(z).
(iii) This is the most delicate part. Unfortunately, we cannot apply Jessen–Wintner the- orem [18, Theorem 13] to fp(z), since ρ0 (in the notation of the latter theorem) depends on p. Therefore, we need to establish the following explicit version of their result.
Lemma 4.5. Let ρ > 0 and let F(z) = X
k≥1
akzk be absolutely convergent for |z| < ρ+ǫ, ǫ > 0. Let S ⊂ C denote the parametric curve {S(θ)}θ∈[0,1] = {F(re2πiθ)}θ∈[0,1]. Let Dr be the distribution on C = R2 defined as the direct image of the normalized Haar measure on the circle of radius r in C by F and let D˜r = FDr be its Fourier transform. Assume that
|a1| 6= 0. Then, if
ρ′′′ = |a1|
√2 X
k≥2
k3|ak|ρk−2
!,
for any r < ρ0 = min(ρ, ρ′′′) we have D˜r(z)≪r,F (1 +|z|)−12.
Proof. Our goal is to make the proof of [18, Theorem 13] explicit in order to be able to estimate ρ0. To do so, we will verify the conditions of [18, Theorem 12] by proceeding in several steps.
15
First of all, we want to ensure that F′(z)6= 0, and the curve S is Jordan. Put ρ′ = |a1|
√2X
k≥2
k|ak|ρk−2.
Ifr <min(ρ, ρ′),we haveF′(z)6= 0 for allz ∈ Dr=B(0, r),andF is injective onDr.Indeed, either|Rea1| or|Ima1|is greater than |a1|
√2.Without loss of generality we can suppose that
|Rea1| ≥ |√a1|
2. Then
|ReF′(z)| ≥ |Rea1| − |z|X
k≥2
k|ak|ρk−2 ≥ |a1|
√2 − |z|X
k≥2
k|ak|ρk−2 >0
on Dr, in particular F′(z) 6= 0. The sign of ReF′(z) does not change as the function is continuous, so once more, without loss of generality, we may assume that ReF′(z) > 0.
Then, for z1 6=z2 two points inDr,we have by convexity of Dr, ReF(z2)−F(z1)
z2−z1
= Z 1
0
ReF′(z1+t(z2−z1))dt >0,
which proves the injectivity. ThusF is a conformal transformation andS is a Jordan curve.
The next step is to get a condition for the curve S to be convex. We use a well-known criterion [29, Part 3, Chapter 3, 108], stating that S is convex if
RezF′′(z) F′(z) >−1 on|z|=r. The estimate
RezF′′(z) F′(z)
≤ |zF′′(z)|
|F′(z)| ≤ |z|P
k≥2k(k−1)|ak|ρk−2
|a1| − |z|P
k≥2k|ak|ρk−2 ≤ |z|P
k≥2k(k−1)|ak|ρk−2
|a1|
1−√12
for r < min(ρ, ρ′) implies that the condition is satisfied once the left-hand side is less than one, that is
r < ρ′′ = |a1|(2−√ 2) 2P
k≥2k(k−1)|ak|ρk−2.
Now, the condition (i) of [18, Theorem 12] is satisfied for all r < ρ.As for (ii) we consider the function
gτ(θ) =X
k≥1
|ak|rkcos 2π(kθ+γk−τ),
where τ ∈ [0,1) is fixed and ak = |ak|e2πiγk. We have to prove that for r explicitly small enough, its second derivative has exactly two roots on [0,1). We compute
hτ(θ) =−gτ′′(θ)
4π2r =|a1|cos 2π(θ+γ1−τ) +rX
k≥2
j2|ak|rk−2cos 2π(kθ+γk−τ), so
h′τ(θ) =−2π|a1|sin 2π(θ+γ1−τ)−2πrX
k≥2
k3|ak|rk−2sin 2π(kθ+γk−τ).
16
